Characterization of fluid-solid phase transition of hard-sphere fluids in cylindrical pore via molecular dynamics simulation

Characterization of fluid-solid phase transition of hard-sphere fluids in cylindrical pore via molecular dynamics simulation

Huan Cong Huang, Sang Kyu Kwak, and Jayant K. Singh

Citation: The Journal of Chemical Physics 130, 164511 (2009); doi: 10.1063/1.3120486 View online: http://dx.doi.org/10.1063/1.3120486

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/130/16?ver=pdfcov Published by the AIP Publishing

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Characterization of fluid-solid phase transition of hard-sphere fluids in cylindrical pore via molecular dynamics simulation

Huan Cong Huang,¹Sang Kyu Kwak,^1,a兲and Jayant K. Singh²

1Division of Chemical and Biomolecular Engineering, School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore 637459, Singapore

2Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

共

Received 26 November 2008; accepted 27 March 2009; published online 27 April 2009

兲

Equation of state and structure of hard-sphere fluids confined in a cylindrical hard pore were investigated at the vicinity of fluid-solid transition via molecular dynamics simulation. By constructing artificial closed-packed structures in a cylindrical pore, we explicitly capture the fluid-solid phase transition and coexistence for the pore diameters from 2.17␴ ^{to 15}␴. There exist some midpore sizes, where the phase coexistence might not exist or not clearly be observable. We found that the axial pressure including coexistence follows oscillatory behavior in different pore sizes; while the pressure tends to decrease toward the bulk value with increasing pore size, the dependence of the varying pressure on the pore size is nonmonotonic due to the substantial change of the alignment of the molecules. The freezing and melting densities corresponding to various pore sizes, which are always found to be lower than those of the bulk system, were accurately obtained with respect to the axial pressure. ©2009 American Institute of Physics.

关DOI:

10.1063/1.3120486兴

I. INTRODUCTION

It has been well recognized that confined fluids in nano- to micrometer pore sizes exhibit different properties and be- haviors compared to those of bulk fluids at the same thermo- dynamic conditions.^1–5 An answer to this phenomenon can be found from understanding confining effects toward the structural arrangement and the movement of molecules, which are much restricted than in the bulk.^6,7In past decades many studies on confinement have been done in typical ap- plications such as porous media, which retain curled and entangled cylindrical pores, and carbon nanotubes

共CNTs兲,

which resemble a rigid and straight cylindrical pore. It has been shown that even with a generalized pore shape, the fundamental knowledge of fluid-pore systems can be cap- tured for realistic systems;^8–10 however, the main focus in these systems has been brought into the state of fluids.

In correspondence with the growing demands for under- standing solid-phase-related problems in cylindrical confine- ment, there exist some studies that chose to use the hard cylindrical wall with hard-sphere model out of many candi- date models due to its theoretical tractability and repulsive hard-core-oriented nature. The former reason is based on several numerical evidences on the existence of the fluid- solid phase transition in both for bulk^11–14 or confined systems¹⁵with the absence of attractive interactions. The lat- ter provides the system under effects of dominant influence of entropy, thus leading one to focus on the relation of the phase transition and the structure of confined particles with varying pore sizes. Moreover, this simple model resembles realistic colloidal particles^16–18 and peapod-related system

共e.g., C

60 encapsulated in CNT兲 that can be described by quasi-one-dimensional approach in statistical mechanics.¹⁹

Mon and Percus²⁰performedNPT

共i.e., constant number

of particles, pressure, and temperature

兲

Monte Carlo

共

MC

兲

simulation of confined hard-sphere fluid to obtain the effect of narrow pore on the equation of state

共i.e., pressure versus

density兲. They found a nonmonotonic change of the equation of state caused from varied structures of confined particles.

Later, theoretical quantification for hard-sphere fluids in the very narrow pore was established.²¹In recent years, Gordillo et al.¹⁵ observed the onset of freezing shown by a sharp change of the density at certain radii of pores via the same NPTMC method and elucidated that the cause of this behav- ior is microstructural transformation. With effects of the at- tractive potential, Koga and Tanaka⁷studied Lennard-Jones

共LJ兲

confined fluids analogous to argon atoms trapped in single-wall CNT via molecular dynamics

共MD兲

method. De- pendence of temperature and spontaneous structures of soft- sphere particles were examined with respect to variation of phases.

The aforementioned works generally focused on a nar- row range of pores, but we wish to extend the idea of pore- size dependence on hard-sphere fluid-solid model with asymptotic increase in pore-size range up to 15␴^{, where}␴^is the diameter of the hard sphere, via collision-based MD method. In particular, MD technique enables one to find the accurate fluid-solid coexistence region provided that the con- figurations are known depending on pore sizes and further to obtain the information of the structural change during the transition. Note that we use the term fluid-solid phase tran- sition or phase transition in place of transition of confined fluids from disordered

共i.e., fluidlike兲

to ordered

共i.e., solid-

like

兲

state in this paper. The rest of the paper is organized as follows: In Sec. II we briefly introduce the fluid and bound-

a兲Author to whom correspondence should be addressed. Electronic mail:

skkwak@ntu.edu.sg.

0021-9606/2009/130共16兲/164511/6/$25.00 130, 164511-1 © 2009 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

ary model in our simulation and the details of simulation conditions in this work. Section III presents the results and discussion followed by conclusion in Sec. IV.

II. SIMULATION MODEL AND DETAILS

In this work, the fluid-fluid interaction is represented by the hard-sphere potential

u_f-f

共r

ij

兲

=

再

^{⬁0,, rrijijⱖ⬍␴␴,,}

冎 ^共1兲

wherer_ijis the distance between two particles. The cylindri- cal pore is then modeled by creating an inscribed cylindrical boundary in a rectangular simulation box so that fluid par- ticles are confined in the cylindrical shape. The cylindrical pore is considered perfectly rigid and has no slip, i.e., when a particle collides at the wall, the normal component of its momentum turns into its opposite direction with the same magnitude while the tangential component is invariant. The fluid-wall interaction is represented by the following equa- tion:

u_f-w

共r兲

=

再

^{⬁,0, rrⱖ⬍0.50.5␴␴,,}

冎 ^共2兲

whereris the distance from the wall boundary to the center of the particle. Periodic boundary condition is applied along the pore axis, which is set to the zdirection

共see Fig.

1兲 in this study.

The units in this work are simulation units, which use amu for mass, Å for length, and ps for time; thus the unit of temperature T^ⴱ is amu Ų/ps² and that of pressure P^ⴱ is amu/Å ps². Since we take unity for hard-sphere diameter␴^, the number density ␳^ⴱ

共i.e., =

␳␴³

兲

becomes dimensionless.

All length scales are in unit of␴. We have performed a series of MD simulations in the microcanonical

共NVE兲

ensemble, whereNis the number of particles,Vis the volume of simu- lation box, andEis the energy. To construct and perform MD simulations,ETOMICA

共Ref.

22兲simulation package was em- ployed. The reduced time step ⌬t^ⴱ was set to 0.01. The re- duced temperatureT^ⴱwas fixed at 1.0, as it has no effects to the transition behavior in the hard-sphere system. For the pore diameter less than 4.0␴ the simulation was conducted with the system size of 1000 particles, and 2000 or more particles were applied for pores larger than 4.0␴ to keep the length of the tube long enough to avoid finite size effects.

Simulations of double system size were also conducted to see that there are no significant difference of properties when the length of the tube is long enough. The total time steps for equilibration were set to 10⁶ at constant temperature and once the system reached equilibrium, another same number of steps was run inNVEensemble to produce data. Note that we omit ␴ in the length unit hereafter. For P_zz^ⴱ

共i.e., axial

pressure兲, we use the pressure tensor equation obtained from collision-based virial theorem.²³Since the system of interest is not under effect of shear stress, we consider only main diagonal terms in the pressure tensor, where P_xx^ⴱ and P_yy^ⴱ

共

e.g., the same magnitude in average

兲

are in the radial direc- tion whileP_zz^ⴱ is in the axial one.

It is generally very difficult to simulate high density sys- tems mainly due to the difficulty of constructing the random and spontaneous systems above the freezing density. The ar- tificially constructed initial packing at high density may af- fect the properties of the system since these packings are not spontaneously constructed. Those may be at the jammed state that could not be changed to other possible configura- tions at the same density. A study of spontaneous close- packed structures of hard-sphere particles in cylindrical shape atT^ⴱ= 0 was done by Pickettet al.,⁶who found differ- ent close-packed configurations of particles in a narrow cy- lindrical shape

共i.e., diameter approximately smaller than

2.2

兲

and the dependence of maximum fraction on pore size.

Another study of close-packed structure of confined LJ sys- tems was conducted by Koga and Tanaka.⁷ Spontaneous close-packed structure of particles in larger pores becomes severely complex and is still an unsolved problem. To obtain a high density, one solution is starting the simulation at ran- dom fluidlike structure and carefully compressing the system to high density using NPT MC method. However, by NPT scheme, compressing the system from low density to high usually leads a system to a jammed state at high density somewhere above melting or to a metastable state for the worse case. This might have been an issue for Gordillo et al.,¹⁵ who did not observe the freezing atD= 2.18, while we found that at 2.17. Also, their simulations ended at P_zz^ⴱ

= 20.0, which is lower than the coexistence pressure we found for D= 2.18. We suspect that they could not perform simulations at higher pressure due to the low probability of volume decrease; the system did not go through chirally or- dered solidification. To this end, we realized that NPT scheme may not be suitable for accurate determination of the

FIG. 1. Screenshots of the simulations at high densities, at which solid structures can be observed. The cross section and side views are shown for different pore sizes forD= 1.7共quasi-one-dimensional system兲, 2.2 and 2.5 共one layer兲, 3.01, 3.1, 3.2, 3.5, 3.6, 3.7, 3.9, 4.0, and 4.5共two layers兲, and 5.0, 6.0, 7.0, 8.0, 10.0, and 12.0共three or multilayers兲.

164511-2 Huang, Kwak, and Singh J. Chem. Phys.130, 164511共2009兲

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fluid-solid transition. Thus we develop our own scheme to construct the initial configurations, which can bear the high density enough to be above melting.

Here we give a brief description of this simplified scheme. In the radial direction, the particles are packed into layers of regular polygons with different shapes. Denote the radius of theith layer within the pore diameter as riand the number of sides of a polygon asSi. We begin with the outer layer. For a pore diameter D, the maximum accessible dis- tance of the center of a particle to the center of the cylinder is

共D

− 1兲/2; then rn=

共D− 1兲

/2 −␦^{, where n} denotes the maximum number of layers and ␦ is a small number

共i.e.,

0.001 or smaller兲, which can be modulated to position the particles radially so that the surface of the particle at the outmost layer does not overlap with the cylindrical wall sur- face. In the next step, we calculate the maximum possible number of sides Sn of a regular polygon, whose length of side is 1.0 and circumradius is smaller thanrn. Then we lay the center of particles into a regular polygon ofSnside and circumradius rn. The

共n

− 1兲th layer is packed with r_n−1=rn

− 1.0−␦ and the same procedure is repeated until the inner- most layer is reached. If the radius of the innermost layerr₁ is not large enough to hold a polygon shape or layer of two atoms, we then pack the particles into a zig-zag structure. In the axial direction, the regular polygons are stacked with the structure ABAB¯, in which Aand B have the same shape, whileBrotates with an angle of␲/S_iagainstA. The distance of A andB is set to 0.87

共i.e., slightly larger than 冑

3/2 to avoid the overlap of particles and assure enough density兲.

Note that this scheme has some shortcomings; first, it does not produce extremely high density

共i.e., near close-packed

density in the bulk兲, and second, at high density, as the pack- ing is not spontaneously constructed, the properties obtained from these initial packing may be specific beyond the melt- ing point. Nevertheless, this compromising solution of pack- ing structure serves our purpose, which predicts the target fluid-solid transition ranges of density and pressure of our interest.

The microscopic structure of the system is monitored by the one-body radial density profile ␳

共

r^ⴱ

兲

and the axial pair distribution function g

共

z^ⴱ

兲

, where r^ⴱ=r/␴ ^{and zⴱ=z}/␴^{, re-} spectively. For the fluid confined in cylindrical pores, the radial density profile␳

共

r^ⴱ

兲

is obtained by dividing the pore into n concentric cylindrical shells with the same thickness

⌬r=

共

D− 1.0

兲

/2nthen sample the number of particles in each shell divided by the shell volume. Thus, the segment of␳

共r

^ⴱ

兲

can be expressed as

␳

共r

i/␴

兲

=

具N

i

典

␲

共共r

i/␴

兲

²−

共r

i−1/␴

兲

²

兲L

,

共3兲

whereNi is the number of particles, whose distances to the pore center are in the ranger_i−1tori, andri=⌬r⫻i.Lis the length of the cylindrical pore. A simple modification from the pair correlation function based on the spherical coordi- nate can lead to the axial pair distribution function, which is extracted from the following equation:

冕

z^ⴱ=0 z^ⴱ=⬁

␳^ⴱg共z^ⴱ

兲

␲

冉

^R␴

冊

^2dzⴱ

^⬇

^N,

^共4兲

whereNis the total number of particles,␳⁼N/Vis the num- ber density, and R is the radius of the cylindrical pore. The upper limit ofZ^ⴱ is changed to the axial length of the simu- lation box, L in calculation. A normalization factor

共i.e.,

N␳␲^R2L兲has been used to present the results in this study.

III. RESULTS AND DISCUSSION

It has been generally considered that a purely one- dimensional system with shot-ranged interactions cannot have a phase transition at finite temperature.²⁴ Hard-sphere particles confined with pore diameters less than 2.0 can be considered as such kind of system, as each particle can only have two and unchangeable neighbors in interaction and the motions of particles are mostly restricted at one dimension, which is the axis of the cylindrical core

共see Fig.

1for snap- shots of confined structures at different pore sizes兲. Figure2 presents the equation of state of the aforementioned system.

We observe that forD= 1.1,P_zz^ⴱ is lower than the bulk value

共

i.e., Carnahan and Starling²⁵and Speedy²⁶

兲

at the same den- sity. This is due to less collision of particles, which can be considered as a major confining effect that comes from the presence of narrow pore size; the frequency of occurrence of intercollision between particles is comparable to that of the particle-wall collision. For other pore sizes

共i.e.,

D= 1.3, 1.5, 1.7, and 1.9兲, the axial pressures almost overlap with the bulk value at low density, and as density increases, they greatly deviate from the bulk value and diverges to the infi- nite at the close-packed structures corresponding to their pore sizes. We particularly observe that the dependence of the axial pressure on the pore size is nonmonotonic. Over densities higher than 0.4, it is clear that the axial pressure increases with the pore diameter from 1.1, reaches its maxi- mum around 1.7, and then drops as the diameter increases to 1.9. This behavior indicates the structural change of the con- fined particles, which can be easily observed by the axial pair distribution function g共z^ⴱ

兲 共not shown兲. We did not observe

any discontinuous points indicating the occurrence of the phase transition in these pore sizes.

FIG. 2. 共Color online兲The axial pressure P_zz^ⴱ vs the average density␳^ⴱof confined hard spheres in the cylindrical hard pore. The pore diameters are in the range from 1.1 to 1.9. The solid line for Carnahan and Starling’s equa- tion of state共Ref.25兲is plotted for comparison. Error is smaller than the size of symbols.

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The axial pressure with respect to the average number density is presented in Fig.3 for the pore sizes ranged from 2.17 to 2.5. Such systems consist of only one cylindrical layer of particles of twisted helices or simple regular polygon shapes—triangle and square

共see Fig.

1兲. Compared to the case Dⱕ1.9, these systems show higher packing over unit volume. The axial pressure is always larger than the bulk value over the density of interest; the intercollision between particles is more rigorous than the particle-wall collision. We obtain the smallest pore size

共i.e.,

D= 2.17兲for observing the phase transition of hard-sphere fluids in the hard cylindrical pore. Due to the severe sensitivity of the pore size on the phase transition, we asymptotically increase the diameter to capture the transition of the fluid-solid coexistence region.

Note that NPT MC method is not easily able to obtain the clear phase transition with the same accuracy. While the pore size gradually increases until 2.2, the axial pressure de- creases in a nonmonotonic manner, but the freezing and melting densities linearly decrease with nearly the same range of the coexistence region

共

i.e., weak dependence on pore size

兲

. We observe that the fluidlike

共

i.e., disordered

兲

structure below the freezing density transforms to the solid- like

共

i.e., series of triangular polygons in the axial direction

兲

through forming a twisted helical structure at coexistence.

Typical pair distribution functions atD= 2.2 are presented to show the confined structures at different phase branches in Fig.4: fluid, around freezing, in coexistence, around melting, and solid branch. Note the appearance of small and irregular shape of peaks among regular peaks at coexistence, which indicates helical shape of the confined particles. Over this range of pore sizes, the confined particles show the same structures except atD= 2.5. The freezing and melting densi- ties shift to high with a shorter coexistence region but with the same pressure as in D= 2.2

共i.e., similar intercollision

numbers兲. We attribute this phenomenon to a structural change, in which indeed the tomogram of the confined par- ticles depicts a square

共see Fig.

1兲.

The axial pressures in the systems of two cylindrical layers of particles are presented in Fig.5. Such systems in- volve the interactions of particles between different layers, thus making the phase-transition behavior much more com- plex. Similar to the one layer case, the pressures in these pores are larger than the bulk value at the same density.

However, one can expect the oscillatory behavior of the equation of state, which may heavily rely on the confined structure of the particles. Observed in Figs. 1 and 6, the outer-layer particles are aligned following loosely elongated helical shape in the axial direction so that the inner-layer particles form a zigzag

共i.e., well fitting of the inner-layer

particles onto the contacting surface of the outer-layer par-

FIG. 4. 共Color兲The axial distribution functiong共z^ⴱ兲, wherez^ⴱ=z/␴^{, atD}

= 2.2. The different lines correspond to the densities at fluid branch共␳^ⴱ

= 0.77兲, around freezing density 共␳^{ⴱ= 0.80}兲, within fluid-solid coexistence 共␳^{ⴱ= 0.85}兲, around melting density共␳^{ⴱ= 0.88}兲, and solid branch共␳^{ⴱ= 0.90}兲.

FIG. 5. 共Color online兲The axial pressure P_zz^ⴱ vs the average density␳^ⴱof confined hard spheres in the cylindrical hard pore. The pore diameters are in the range from 3.01 to 4.5. The solid lines are the same as shown in Fig.3.

Error is smaller than the size of symbols at fluid and solid phases but twice that of the symbols at coexistence.

FIG. 3. 共Color online兲The axial pressurePzzⴱ vs the average density␳^ⴱof confined hard spheres in the cylindrical hard pore. The pore diameters are in the range from 2.17 to 2.5. The solid lines in different phases are obtained from the equation of states for fluids and solids共i.e., face-centered-cubic hard-sphere crystal兲 from Carnahan and Staring 共Ref. 25兲 and Speedy 共Ref.26兲, respectively. Error is smaller than the size of symbols.

FIG. 6. 共Color online兲Radial density profiles␳共r^ⴱ兲␴^{3, whererⴱ=r}/␴^{, for} the pores in the range fromD= 3.01 to 4.5. The profiles show the different positions formed by the two layers of particles within the pores.

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ticles

兲

atD= 3.01, the outer layer is recoiled to form a more compact helical shape, which makes the inner-layer particles aligned in a line atD= 3.1, and then the hexagonal shape of the outer-layer particles is formed throughout the axial direc- tion with the inner-layer particles formed in a line at D

= 3.2. Indeed, we found the oscillatory values of coexistence pressure and densities. ForD= 3.01, we observed a wide co- existence region

共i.e., from

␳^ⴱ

⬇

0.870 to 0.935, indicating a large shift toward the bulk phase-transition region, compared to the case of D= 2.5兲and the coexistence pressure is about 11.9, which is quite close to the coexistence pressure in the bulk. A little increase in the diameter to 3.1 largely shifts the coexistence region

共i.e., from

␳^ⴱ

⬇

0.950 to 0.991兲 and the coexistence pressure is around 20.2, but atD= 3.2, the coex- istence region drops back again

共i.e., from

␳^ⴱ

⬇

0.863 to 0.892兲with the pressure at around 11.1. While the pore size continuously increases, we found that the system exhibits the oscillatory behavior of the equation of state with the narrow phase-transition region. In particular, over the pore range from 3.5 to 4.5, the largest coexistence pressure was found at D= 3.6 due to a probable occurrence of the closed-packed structure as similarly found inD= 3.1. Most interestingly, the phase transition disappears atD= 3.7. We speculate that the system is the jammed structure, which can be indirectly seen from very low density for the inner layer and high one for the outer layer, which forms a concrete octagon shape. We have also not seen clear phase transitions atD= 3.9 and 4.0.

For the multilayer systems with three or more layers, the pressure is close to the bulk value of the fluid phase and the phase transition generally happens below the freezing den- sity of the bulk system. The equations of states are presented in Fig.7and typical axial pair distribution functions for fluid and solid are presented in Fig. 8 to show the relative struc- tures of the confined particles between small

共D= 3.01 and

3.5兲 and large

共D= 6.0兲

pores. Fluid-solid coexistences are clearly observed for Dⱖ6.0 while those are absent

共or not

clear兲 in D= 5.0 and 5.5. The freezing and melting points shift to higher densities and the coexistence pressure rises as the pore size increases from D= 6.0 to 8.0. For Dⱖ10, we found that the coexistence pressure is very close to the bulk value but with a shorter range of the coexisting region. We do not observe a strong oscillatory behavior of the equation

of state over this range of the pore sizes indicating less effect of the different packing structures. The values of the freezing and melting densities are presented in TableIwith respect to the coexistence axial pressures depending on the pore sizes at the phase transitions. In these large pores, each layer could be regarded as a hexagonal plane rolling into a cylindrical pore in general. As the pore size increases, atoms are less directly affected by the wall boundary so that the bulk struc- ture might be locally formed in the vicinity of the cylinder axis; however, we do not specify a notable threshold to dis- tinguish whether the particles follow the behavior of the bulk system or that of the highly confined one. Lastly, we provide relative axial pressures with respect to those at bulk depend- ing on pore sizes when the density of the system is below 0.7 in Fig.9. At very low densities from 0.1 to 0.2, the pressures monotonically reduce toward those at bulk as pore sizes in- crease. As density increases, we observed that the range of pore sizes for the nonmonotonic behavior of the relative

FIG. 7. 共Color online兲The axial pressureP_zz^ⴱ vs the average density␳^ⴱof confined hard spheres in the cylindrical hard pore. The pore diameters are in the range from 5.0 to 15.0. The bulk lines are the same as Figs.3and5.

Error is smaller than the size of symbols at fluid and solid phases but twice that of the symbols at coexistence.

FIG. 8.共Color兲The axial pair distribution functiong共z^ⴱ兲, wherez^ⴱ=z/␴, for the poresD= 3.01, 3.5, and 6.0 at fluid branch and solid branch, respectively.

TABLE I. Estimated data of densities at freezing共␳ⴱf兲and melting共␳mⴱ兲and coexistence axial pressure共P_zz,coe^ⴱ 兲for various pore sizes. The coexistence pressure is calculated by averaging the values in the coexistence region. The freezing and melting densities are then estimated by fitting the second-order polynomials to fluid and solid branches, respectively. The values shown in parentheses represent errors in the last digits.

Diameter ␳ⴱf ␳_m^{* P}zz,coeⴱ

2.17 0.852共1兲 0.935共1兲 26.1共2兲 2.18 0.832共1兲 0.914共2兲 20.6共1兲 2.19 0.813共2兲 0.895共4兲 17.2共3兲

2.2 0.803共1兲 0.878共2兲 14.91共9兲

2.5 0.826共2兲 0.893共3兲 14.8共4兲

3.01 0.870共4兲 0.935共3兲 11.9共2兲

3.1 0.950共1兲 0.991共2兲 20.2共4兲

3.2 0.863共2兲 0.892共1兲 11.1共1兲

3.5 0.852共2兲 0.893共2兲 13.9共2兲

3.6 0.870共1兲 0.909共1兲 14.8共1兲

4.5 0.897共5兲 0.929共7兲 12.9共6兲

6.0 0.858共2兲 0.879共6兲 9.73共6兲

7.0 0.865共3兲 0.899共4兲 9.8共1兲

8.0 0.883共3兲 0.911共6兲 10.4共1兲

10.0 0.903共4兲 0.947共8兲 11.0共2兲

12.0 0.922共1兲 0.983共9兲 11.7共4兲

15.0 0.922共4兲 0.961共3兲 11.5共1兲

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pressure becomes larger and the pressures cross over to those at lower densities. This phenomenon indicates that the con- siderable structural rearrangements started even at liquidlike densities. The nonmonotonic behavior of the relative pres- sure spreads over larger pore sizes as the density increases;

thus complex crossover of the equation of state happens at coexisting and solidlike regions as shown in this work.

IV. CONCLUSION

In this study, we consider cylindrically confined systems under entropic dominance to capture the effects of confine- ment on the phase transition of disordered particles. In par- ticular, we have determined the equation of state of hard spheres confined in a cylindrical hard pore with a wide range of pore sizes viaNVEMD method. Besides agreement with others^15,20,21 in that the behaviors of the equation of states highly depend on the pore size when it is narrow, we found that a little variation of the pore diameter dramatically changes the freezing and melting process; large shifts of the freezing and melting densities as well as the coexistence pressure and the degree of property dependence on the pore size become weaker as it increases. We did not observe the phase transition in the pore diameters less than 2 while the transition is observed for the pore diameters larger than 2.16, also suggested by Gordilloet al.

共i.e., 1 + 2 冑

3/3兲.¹⁵ We pre- sented the detailed data for the coexistence of fluid-solid phase and tracked the structural change during the onset of

the fluid-solid transition. Our results predict that all proper- ties found generally do not contain simple monotonic depen- dences on the pore size mainly due to the diversity of hard- sphere packing structures with respect to different pore sizes.

Indeed, certain pore sizes make the systems without any clear phase transitions, but one can consider other spontane- ous packing structures, which we leave for a future work.

ACKNOWLEDGMENTS

This work was supported by Nanyang Technological University

共Grant Nos. M58120005-SUG and M52120043-

RG39/06兲. Computational resources have been provided by the School of Chemical and Biomedical Engineering.

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